1. Given n € N, define F, (the nth Fibonacci Number) as Fo= 0, F₁ = 1, FnFn-1 + Fn-2 for all n ≥ 2. Prove that F₁ < 2" for all n € N. (Hint: If a

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Q: * 30. Let n and k be integers with 1 ≤ k ≤n. Show that n Σ() ( k=1 k n - 1) = (2n+2)/2 - (2n).

A: The answer for the given questions has been solve step by step below please find the attachments…

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Q: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

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Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

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Q: 12. Let the "Fibonacci-2" numbers g, be defined as follows: 91 = 2, 92 = 2, and gn+2 =( In+1)( In)…

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Q: Define the Fibonacci numbers as Fo = 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1…

A: Let's prove the identity Fn+1Fn−1−Fn2=(−1)n for n≥1 using mathematical induction.Base Case (n =…